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A001685 a(0) = 1, a(1) = 2, a(2) = 3; for n >= 3, a(n) = a(n-2) + a(n-1)*Product_{i=1..n-3} a(i).
(Formerly M0740 N0278)
11
1, 2, 3, 5, 13, 83, 2503, 976253, 31601312113, 2560404986164794683, 202523113189037952478722304798003, 506227391211661106785411233681995783881012463859772443053 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From a continued fraction.

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

John Cerkan, Table of n, a(n) for n = 0..16

V. C. Harris, Another proof of the infinitude of primes, Amer. Math. Monthly, 63 (1956), 711.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012

FORMULA

a(n) = a(n-2) + a(n-1)*a(n-3)*(a(n-1)-a(n-3))/a(n-2). - Vaclav Kotesovec, May 21 2015

a(n) ~ c^(d^n), where d = A109134 = 1.754877666246692760049508896358528691894606617772793143989283970646... is the root of the equation d*(d-1)^2 = 1, c = 1.3081335128180696870655208993764956995000211962454918672885690026423582299... . - Vaclav Kotesovec, May 21 2015

MATHEMATICA

Clear[a]; a[0]=1; a[1]=2; a[2]=3; a[n_]:=a[n] = a[n-2] + a[n-1]*Product[a[j], {j, 1, n-3}]; Table[a[n], {n, 0, 15}] (* Vaclav Kotesovec, May 21 2015 *)

Clear[a]; RecurrenceTable[{a[n]==a[n-2]+a[n-1]*a[n-3]*(a[n-1]-a[n-3])/a[n-2], a[0]==1, a[1]==2, a[2]==3}, a, {n, 0, 15}] (* Vaclav Kotesovec, May 21 2015 *)

PROG

(PARI) a(n)=if(n<3, max(0, n+1), a(n-2)+a(n-1)*prod(i=1, n-3, a(i))) /* Michael Somos, Feb 01 2004 */

CROSSREFS

Cf. A003686, A064526, A109134.

Sequence in context: A056162 A265785 A326372 * A074691 A139589 A152114

Adjacent sequences:  A001682 A001683 A001684 * A001686 A001687 A001688

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by N. J. A. Sloane, Jun 12 2006

STATUS

approved

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Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)